On the exact electric and magnetic ﬁelds of an electric dipole
W.J.M.KortKamp
a
and C.Farina
b
Instituto de Fisica,Universidade Federal do Rio de Janeiro,Rio de Janeiro 21945970,Brazil
Received 3 May 2010;accepted 24 August 2010
We make a multipole expansion directly in Jeﬁmenko’s equations to obtain the exact expressions for
the electric and magnetic ﬁelds of an electric dipole with an arbitrary time dependence.Some
comments are made about the usual derivations in most undergraduate and graduate textbooks and
in literature.©
2011 American Association of Physics Teachers.
DOI:10.1119/1.3488989
The problemof ﬁnding an analytic expression for the elec
tric ﬁeld of a localized but arbitrary static charge distribution
is quite involved.Due to the difﬁculty in obtaining exact
solutions,numerical methods and approximate theoretical
methods have been developed.One of the most important
examples of the latter is the multipole expansion method.For
the origin inside the distribution,the multipole expansion
method gives the ﬁeld outside the distribution as a superpo
sition of ﬁelds,each of which can be interpreted as the elec
trostatic ﬁeld of a multipole located at the origin see,for
instance,Ref.1.The ﬁrst three terms of the multipole ex
pansion correspond to the ﬁelds of a monopole,a dipole,and
a quadrupole,respectively.For an arbitrary distribution with
a vanishing total charge,the ﬁrst term in the multipole ex
pansion is the ﬁeld of the electric dipole of the distribution.
This property is an example of the important role played by
the ﬁeld of an electric dipole.
For localized charge and current distributions with arbi
trary time dependence,the multipole expansion method is a
powerful technique for calculating the electromagnetic ﬁelds
of the system.In the study of radiation ﬁelds of simple sys
tems,such as antennas,it is common to make a multipole
expansion and to keep only the ﬁrst few terms.It is worth
emphasizing that in contrast to what happens in the case of
static charge distributions,the leading contribution to the ra
diation ﬁelds comes from the electric dipole term since there
is no monopole radiation due to charge conservation.
The electric ﬁeld of an electric dipole has three
contributions—the static zone contribution proportional to
1/r
3
,the intermediate zone contribution proportional to 1/r
2
,
and the far zone or radiation contribution proportional to 1/r,
where r is the distance from the origin.As we shall see,the
corresponding magnetic ﬁeld has only two contributions. A
general derivation of the exact electric and magnetic ﬁelds of
an electric dipole is lacking from most textbooks.
2–12
In
some cases,the textbooks obtain the exact electromagnetic
ﬁelds for the electric dipole by assuming a harmonic time
dependence for the sources
2–5
or by assuming a particular
model for the charge and current distributions.
6–8
In other
cases,the textbooks are interested only in the radiation
ﬁelds.
9–12
A general derivation of the exact electric and magnetic
ﬁelds of an electric dipole with arbitrary time dependence
can be found in Ref.13.Heras ﬁrst presented a new version
of Jeﬁmenko’s equations in a material medium
14
with polar
ization P and magnetization M and then obtained the exact
dipole ﬁelds.
The purpose of this note is to provide a direct derivation of
the exact electric and magnetic ﬁelds of an electric dipole
with arbitrary time dependence without making any assump
tions for the localized charge and current distributions of
the system.Our starting point will be Jeﬁmenko’s equations
and our procedure consists of making a multipole expansion
up to the desired order.Our procedure does not require Jeﬁ
menko’s equation in matter,which makes our derivation a
very simple one.
Jeﬁmenko’s equations for arbitrary but localized sources
are given by
15–19
Er,t =
1
4
0
dr
r
,t
R
R
3
+
dr
˙
r
,t
R
cR
2
−
dr
J
˙
r
,t
c
2
R
,1
Br,t =
0
4
dr
Jr
,t
R
R
3
+
dr
J
˙
r
,t
R
cR
2
,2
where the integrals are over all space,R=R =r−r
,the
overdot means time derivative for example,
˙
r
,t
=r
,t
/t
,and the brackets … mean that the quanti
ties inside them must be evaluated at the retarded time t
=t −R/c.
We make a multipole expansion,which consists of making
an expansion in powers of r
/r,which is valid outside the
charge and current distributions.Our ﬁnal result will be valid
only outside the sources.Because we are interested in the
dipole ﬁelds,we shall retain only terms up to linear order in
r
.Therefore,we use the following Taylor expansions:
1
R
1
r
+
r
ˆ
r
r
2
,3
R
R
2
r
ˆ
r
+
2r
ˆ
r
r
ˆ
r
2
−
r
r
2
,4
R
R
3
r
ˆ
r
2
+
3r
ˆ
r
r
ˆ
r
3
−
r
r
3
,5
111 111Am.J.Phys.79 1,January 2011 http://aapt.org/ajp © 2011 American Association of Physics Teachers
t −
R
c
t
0
+
r
ˆ
r
c
,6
where t
0
=t −r/c is the retarded time with respect to the ori
gin.Analogously,we expand the source terms about t
=t
0
and keep terms only up to linear order in r
.The results are
r
,t
=r
,t − R/c
r
,t
0
+ r
ˆ
r
/c r
,t
0
+
r
ˆ
r
c
˙
r
,t
0
,
7
Jr
,t
= Jr
,t − R/c
Jr
,t
0
+ r
ˆ
r
/c Jr
,t
0
+
r
ˆ
r
c
J
˙
r
,t
0
,
8
where we used Eq.6.We can make analogous expansions
to obtain approximate expressions for the time derivatives of
the sources,namely,
˙
r
,t
˙
r
,t
0
+
r
ˆ
r
c
¨
r
,t
0
,9
J
˙
r
,t
J
˙
r
,t
0
+
r
ˆ
r
c
J
¨
r
,t
0
.10
For convenience,we denote the three integrals that appear on
the righthand side of Eq.1 by I
1
E
,I
2
E
,and I
3
E
.Similarly,
we denote by I
1
B
and I
2
B
the two integrals that appear on the
righthand side of Eq.2.
We must calculate these integrals up to the desired order.
We ﬁrst consider I
1
E
,substitute Eqs.5 and 7 into the
expression for I
1
E
,and obtain
I
1
E
dr
r
,t
0
+
r
ˆ
r
c
˙
r
,t
0
r
ˆ
r
2
+
3r
ˆ
r
r
ˆ
r
3
−
r
r
3
11
r
ˆ
r
2
dr
r
,t
0
+
3r
ˆ
r
3
r
ˆ
dr
r
,t
0
r
−
1
r
3
dr
r
,t
0
r
+
r
ˆ
cr
2
r
ˆ
dr
˙
r
,t
0
r
,
12
where quadratic terms in r
are neglected.The integral in the
ﬁrst termon the righthand side of Eq.12 is the total charge
of the distribution,which is independent of time due to
charge conservation.This term corresponds to the monopole
termof the multipole expansion because it is the ﬁeld created
by a point charge Qr
,t
0
dr
ﬁxed at the origin.This
term does not contribute to the dipole ﬁeld.We can identify
the electric dipole moment of the distribution at time t
0
in the
next two integrals on the righthand side of Eq.12,namely,
pt
0
=dr
r
,t
0
r
.After we interchange the time deriva
tive with the integration in the last integral on the righthand
side of Eq.12,we can identify the time derivative of the
electric dipole moment of the distribution at time t
0
,namely,
p
˙
t
0
=d/dtdr
r
,t
0
r
.Hence,I
1
E
is given by
I
1
E
Qr
ˆ
r
2
+
3r
ˆ
pt
0
r
ˆ
− pt
0
r
3
+
r
ˆ
p
˙
t
0
cr
2
.13
We next use an analogous procedure and substitute into
the expression for I
2
E
the approximations in Eqs.4 and 9.
The result is
I
2
E
2r
ˆ
p
˙
t
0
r
ˆ
− p
˙
t
0
cr
2
+
r
ˆ
p
¨
t
0
r
ˆ
c
2
r
.14
To calculate I
3
E
,I
1
B
,and I
2
B
,it is convenient to write the
Cartesian basis vectors as e
ˆ
i
=
x
i
i =1,2,3 so that any
vector b can be written in the form b=b
i
e
ˆ
i
=b e
ˆ
i
e
ˆ
i
=b
x
i
e
ˆ
i
,where the Einstein convention of summation
over repeated indices is assumed.As it will become clear,it
sufﬁces to make the approximations R→r and J
˙
r
,t
→J
˙
r
,t
0
in the integrand.Hence,we have
I
3
E
−
1
c
2
r
dr
J
˙
r
,t
0
= −
1
c
2
r
e
ˆ
i
dr
ˆJ
˙
r
,t
0
x
i
‰ 15a
=
1
c
2
r
e
ˆ
i
dr
x
i
ˆ
J
˙
r
,t
0
‰
= −
1
c
2
r
e
ˆ
i
dr
x
i
¨
r
,t
0
,15b
where in passing from Eq.15a to Eq.15b we integrated
by parts and discarded the surface term because the integra
tion is over all space and the sources are localized.In the last
step we used the continuity equation.The second time de
rivative of the electric dipole moment of the distribution at
time t
0
appears in Eq.14,and thus we obtain
I
3
E
−
p
¨
t
0
c
2
r
.16
We now see why it was not necessary to go beyond zeroth
order in the expansion of J
˙
r
,t
in the calculation of I
3
E
.
The electric dipole moment of the distribution is already ﬁrst
order in r
so that the next order termwould contribute to the
next terms of the multipole expansion,namely,the magnetic
dipole term and the electric quadrupole term.
The integrals I
1
B
and I
2
B
will give the contribution to the
magnetic ﬁeld of the electric dipole term.We proceed as we
did for I
3
E
,namely,it sufﬁces to use only the zeroth order
approximation for the expansions of the current and its time
derivative.We have
112 112Am.J.Phys.,Vol.79,No.1,January 2011 W.J.M.KortKamp and C.Farina
I
1
B
−
r
ˆ
r
2
dr
Jr
,t
0
17a
=−
r
ˆ
r
2
e
ˆ
i
dr
ˆJr
,t
0
x
i
‰ 17b
=
r
ˆ
r
2
e
ˆ
i
dr
x
i
ˆ
Jr
,t
0
‰ 17c
=−
r
ˆ
r
2
e
ˆ
i
dr
x
i
˙
r
,t
0
=
p
˙
t
0
r
ˆ
r
2
,17d
where in the last step we identiﬁed the time derivative of the
electric dipole moment of the distribution.As before,it is
straightforward to show that
I
2
B
p
¨
t
0
r
ˆ
cr
.18
We substitute the results in Eqs.13,14,and 16 into
Eq.1,discard the monopole term,substitute Eqs.17d and
18 into Eq.2,and obtain the exact electric and magnetic
ﬁelds associated with the electric dipole term of an arbitrary
localized distribution of charges and currents,
Er,t =
1
4
0
3r
ˆ
pt
0
r
ˆ
− pt
0
r
3
+
3r
ˆ
p
˙
t
0
r
ˆ
− p
˙
t
0
cr
2
+
r
ˆ
r
ˆ
p
¨
t
0
c
2
r
,19
Br,t =
0
4
p
˙
t
0
r
ˆ
r
2
+
p
¨
t
0
r
ˆ
cr
,20
which are the desired ﬁelds.Equation 19 is given,but not
derived,in Ref.20.
The last terms on the righthand side of Eqs.19 and 20
are proportional to 1/r and correspond to the radiation ﬁelds
of an electric dipole.For the idealized case of a point electric
dipole ﬁxed at the origin,Eqs.19 and 20 are exact for
every point in space,except the origin,where the ﬁelds are
singular.Because we are interested in the ﬁelds outside the
dipole,we shall not be concerned with these singular terms.
A discussion of these terms for a static point electric dipole
can be found in Ref.2,Chap.3.
As an important special case,we consider an electric di
pole with harmonic time dependence.We substitute pt
=p
0
e
−it
into Eqs.19 and 20 and obtain the well known
results,
2
Er,t =
e
−it
4
0
k
2
r
ˆ
p
0
r
ˆ
e
ikr
r
+ 3r
ˆ
p
0
r
ˆ
− p
0
1
r
3
−
ik
r
2
e
ikr
,21
Br,t =
0
e
−it
4
ck
2
r
ˆ
p
0
e
ikr
r
1 −
1
ikr
,22
where k=/c.In Eq.19 note the presence of a term pro
portional to r
−3
and pt
0
,which is a characteristic of the
ﬁeld of a static electric dipole,except that here the electric
dipole moment is evaluated at the retarded time t
0
.This term
dominates in the near zone,where dr,where d is a
typical length scale of the source and is the wavelength of
the electromagnetic ﬁeld.Note the absence of such a term in
Eq.20,which is expected because a static electric dipole
does not create a magnetic ﬁeld.In the radiation zone,where
dr,the dominant terms are proportional to r
−1
and to
the second time derivative of the electric dipole moment.The
transverse character of the radiation ﬁelds is evident.
The results in Eqs.19 and 20 could have been obtained
if,instead of Jeﬁmenko’s Eq.1 for the electric ﬁeld,we had
used the equivalent Panofsky–Phillips equation for the elec
tric ﬁeld
11
given by
Er,t =
1
4
0
dr
r
,t
R
ˆ
R
2
+
dr
Jr
,t
R
ˆ
R
ˆ
+ Jr
,t
R
ˆ
R
ˆ
cR
2
+
1
4
0
dr
J
˙
r
,t
R
ˆ
R
ˆ
c
2
R
.23
Equation 23 is equivalent to Eq.1,as it was shown ex
plicitly in Ref.21,but note that Eq.23 is more convenient
as a starting point for calculating multipole radiation ﬁelds.
22
All multipole radiation ﬁelds can be obtained from the last
terms of the righthand side of Eqs.2 and 23.The dipole
radiation ﬁelds in Eqs.19 and 20 are just the ﬁrst order
contributions of the multipole expansion for the radiation
ﬁelds.The next order contributions,namely,the radiation
ﬁelds of a magnetic dipole and an electric quadrupole,may
be similarly obtained without much more effort.
22
We leave for the interested reader the problem of obtain
ing the exact electric ﬁeld of an electric dipole with arbitrary
time dependence from Eq.23 instead of Eq.1.The exact
electric and magnetic ﬁelds of higher multipoles such as the
magnetic dipole and electric quadrupole can also be obtained
following our approach.In this case,higher order terms in r
must be kept.
We have calculated the exact electric and magnetic ﬁelds
of an electric dipole with arbitrary time dependence,in con
trast to the usual calculations where a harmonic time depen
dence is assumed.Most textbooks use the electromagnetic
potentials.We have instead implemented a multipole expan
sion directly from Jeﬁmenko’s equations.Besides the sim
plicity of our method,it increases the number of problems
that can be attacked directly by using Jeﬁmenko’s equations
see also Refs.23–28.
113 113Am.J.Phys.,Vol.79,No.1,January 2011 W.J.M.KortKamp and C.Farina
ACKNOWLEDGMENTS
The authors are indebted to Professor C.Sigaud for read
ing the paper.They also would like to thank CNPq for partial
ﬁnancial support.
a
Electronic mail:kortkamp@if.ufrj.br
b
Electronic mail:farina@if.ufrj.br
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114 114Am.J.Phys.,Vol.79,No.1,January 2011 W.J.M.KortKamp and C.Farina
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